So in finding the closure of any subset $\mathbb{Z}$, we have the problem:
Given the topological base $\bigl\{[-n,n]:n>0\bigr\}$, find the closure with respect to this topological basis.
I have the solution, but the problem is I do not understand this solution in my notes and it goes:
$k$ will have the closure of $B$ iff $|k| \le n \implies B\cap[-n,n] \ne \varnothing$, hence the closure of $B = [-a,a]$, where $a$ is the maximum of the absolute values of the numbers in $B$.
Note that $[m,n]=\{k\in \mathbb{Z}: m \le k \le n\}$.
I generally do not get this solution at all. Can anyone help me understand it better?
The solution is incorrect. My first hint was this: "the maximum of the absolute values of $B$." Since $B$ may be an infinite subset of $\Bbb Z,$ such maximum may not exist!
The open sets in the given topology turn out to be $\emptyset,$ $\Bbb Z,$ and the elements of the given basis. Consequently, the closed sets with respect to the given topology are $\Bbb Z,$ $\emptyset,$ and the sets of the form $(-\infty,-n-1]\cup[n+1,\infty)$ for positive integers $n.$
Now, take any $B\subseteq\Bbb Z.$ If $B\cap\{-1,0,1\}\neq\emptyset,$ then the closure of $B$ is $\Bbb Z.$ If $B=\emptyset,$ then the closure of $B$ is $\emptyset.$ Suppose, instead, that $B$ is non-empty, and that $|k|>1$ for all $k\in B.$ Take $a=\min\bigl\{|k|:k\in B\bigr\},$ and let $C=(-\infty,-a]\cup[a,\infty).$ I claim that $C$ is the desired closure of $B,$ which is readily shown, as $C$ is closed with respect to the given topology, and as $B\subseteq C.$